Residence times of a Brownian particle with temporal heterogeneity

Home

Search

Collections

Journals

About

Contact us

My IOPscience

Residence times of a Brownian particle with temporal heterogeneity

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 J. Phys. A: Math. Theor. 50 195001 (http://iopscience.iop.org/1751-8121/50/19/195001) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 155.101.97.26 This content was downloaded on 14/04/2017 at 17:53 Please note that terms and conditions apply.

You may also be interested in: Escape from a potential well with a randomly switching boundary Paul C Bressloff and Sean D Lawley Stochastic switching in biology: from genotype to phenotype Paul C Bressloff Moment equations for a piecewise deterministic PDE Paul C Bressloff and Sean D Lawley 100 years after Smoluchowski: stochastic processes in cell biology D Holcman and Z Schuss Time scale of diffusion in molecular and cellular biology D Holcman and Z Schuss Mean joint residence time of two Brownian particles in a sphere O Bénichou, M Coppey, J Klafter et al. Narrow escape for a stochastically gated Brownian ligand Jürgen Reingruber and David Holcman Diffusion on a tree with stochastically gated nodes Paul C Bressloff and Sean D Lawley Joint residence time of N independent 2D Brownian motions O Bénichou, M Coppey, J Klafter et al.

Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 50 (2017) 195001 (18pp)

https://doi.org/10.1088/1751-8121/aa692a

Residence times of a Brownian particle with temporal heterogeneity Paul C Bressloff and Sean D Lawley Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, United States of America E-mail: [email protected] and [email protected] Received 5 January 2017, revised 12 March 2017 Accepted for publication 24 March 2017 Published 6 April 2017 Abstract

We consider a diffusing particle that randomly switches conformational state. Motivated by various scenarios in cell biology, we suppose that (a) the diffusion coefficient depends on the conformational state and/or (b) the particle can only pass through a series of gates in the domain when it is in a particular conformational state. We develop probabilistic methods to analyze this case of diffusion with temporal heterogeneity, and use these methods to calculate the expected residence time in portions of the domain before absorption at a boundary. We find several new phenomena not seen in recent studies of diffusion with spatial heterogeneity, some of which are counterintuitive. In particular, the expected residence times can be non-monotonic functions of (i) the initial distance from the absorbing boundary and (ii) the diffusion coefficients. We focus on one-dimensional intervals, but show how the analysis can be extended to spherically symmetric d-dimensional domains. Keywords: Brownian motion, residence times, stochastic gates, first passage times, temporal heterogenieity (Some figures may appear in colour only in the online journal) 1. Introduction A fundamental quantity in the mathematical theory of random walks and diffusion processes is the occupation time [16, 18], which was originally defined as the time spent by a Brownian particle in R+ = [0, ∞) within a time window of size t. That is, given the Brownian motion X(t) ∈ R , the occupation time T is t T := Θ(X(τ ))dτ , (1.1) 0

1751-8121/17/195001+18$33.00 © 2017 IOP Publishing Ltd Printed in the UK

1

P C Bressloff and S D Lawley

J. Phys. A: Math. Theor. 50 (2017) 195001

where Θ(X) denotes the Heaviside function. The occupation time T is an example of a Brownian functional. Since X(t), t 0, is a Wiener process, it follows that each realization of a Brownian path will typically yield a different value of T, which means that T will be distributed according to some probability density P(T, t|x0 , 0) for X(0) = x0. The statistical properties of a Brownian functional can be analyzed using path integrals, and leads to the well-known Feynman–Kac formula [17]. For a general review of Brownian functionals and their applications, see [19]. An immediate generalization of equation (1.1) is to take t T := IV (X(τ ))dτ , (1.2) 0

d

where X(t) ∈ R is a continuous stochastic process and IV(x) denotes the indicator function of the set V ⊂ Rd , that is, IV(x) = 1 if x ∈ V and is zero otherwise. (Note that for onedimensional (1D) motion, Θ(x) = IR+ (x).) More recently, occupation times have figured prominently in a variety of physical applications under the alternative name of residence times. Examples include the non-equilibrium dynamics of coarsening systems [11, 20], ergodicity properties of anomalous diffusion [10, 21], simple models of blinking quantum dots [22], fluorescent imaging [1], and branching processes [12]. Since a residence time concerns the amount of time that a Brownian particle spends in some bounded or partially bounded domain M ⊂ Rd , a natural extension is to replace the upper limit t by a stopping time such as the first passage time (FPT) to reach a section of the boundary ∂M. This type of residence time has recently played an important role in the calculation of mean first-passage times (MFPTs) in spatially heterogeneous media [9, 23, 24]. In this paper we use probabilistic methods (conditional expectations and the strong Markov property) to determine the stopped residence times of a Brownian particle in a bounded domain with temporal rather than spatial heterogeneity. The introduction of temporal heterogeneity is motivated by the idea that macromolecules in cell biology often switch between different conformational states [2]. For simplicity, we will assume that a particle can randomly switch between two conformational states labeled n = 0, 1 and that this switch has two possible effects: (i) the diffusion coefficient depends on the state n and (ii) there are pores separating different spatial domains and the particle can only pass through a pore when in the state n = 0, say. Thus the pore acts like a stochastic gap junction [6, 7]. These two cases are illustrated in figure 1. The analysis of residence times with a switching diffusion coefficient is presented in section 2, and the extension to stochastically-gated residence times is presented in section 3. In particular, we show how temporal heterogeneity can lead to counterintuitive behaviors, such as the non-monotonic dependence of expected residence times on the initial distance from an absorbing boundary and on the diffusion coefficient. 2. Residence times without gating 2.1. Brownian particle with temporal heterogeneity

Consider a Brownian particle diffusing in the one-dimensional (1D) domain of length L shown in figure 2. The domain is partitioned into cells of size l, ml = L, with a pore or gate at each junction x = kl, k = 1, . . . (m − 1)l . Suppose that the particle switches between two conformational states labelled n = 0,1 such that n(t) ∈ {0, 1} evolves according to a two-state β

Markov chain, 0 1, with the matrix generator α

2



Figure 1. Schematic diagram of two possible trajectories for a Brownian particle that

randomly switches between two conformational states n = 0, 1 according to a twostate Markov chain with transition rates α, β (temporal heterogeneity). (a) Switching between different diffusion coefficients D0 , D1. (b) Brownian particle can only pass through a pore when in the state n = 0.

Figure 2. One-dimensional domain of length L partitioned into m cells of size l with gap junctions at the interior points x = ak = kl, k = 1, m − 1.

−β α A = . (2.1) β −α

We assume that the two conformational states have distinct diffusion coefficients Dn, n = 0, 1 as illustrated in figure 1(a). We then distinguish between two scenarios. (i) Ungated: the particle can pass through the pores in both conformational states so the cell junctions have no effect. (ii) Gated: the particle can only pass through a pore in conformational state n = 0, see figure 1(b).

In this section we focus on the ungated case, and consider the effects of gating in section 3. Let X(t) be the position of the particle at time t, which evolves according to the piecewise stochastic differential equation (SDE) dX(t) = 2Dn dW(t), (2.2)

when n(t) = n ∈ {0, 1}. Here W(t) is a Wiener process with dW(t) = 0 and dW(t)dW(t ) = δ(t − t )dtdt. 3



Assuming the initial conditions X(0) = x0 , n(0) = n0 , we introduce the probability density pn (x, t|x0 , n0 , 0) with P{X(t) ∈ (x, x + dx), n(t) = n|x0 , n0 } = pn (x, t|x0 , n0 , 0)dx.

It follows that pn evolves according to the forward differential CK equation (dropping the explicit dependence on initial conditions) [2, 14] ∂pn ∂ 2 pn (x, t) = Dn + Anm pm (x, t), n = 0, 1. (2.3) ∂t ∂x2 m=0,1

Now suppose that there is an absorbing boundary condition at x = 0 and a reflecting boundary condition at x = L: ∂pn (L, t) (2.4) pn (0, t) = 0, = 0. ∂x Given the first passage time T := inf{t > 0 : X(t) = 0}, (2.5)

we define the (stopped) residence time in the interval (ak , ak+1 ) according to T (2.6) Tk := I(ak ,ak+1 ) (X(t))dt, k = 0, . . . , m − 1. 0

m−1

Note that k=0 Tk = T almost surely. In this paper we are interested in calculating the mean residence times τkm (x0 ), where m τ(2.7) k (x0 ) = Ex0 ,m [Tk ],

with Tk the residence time in the interval (ak , ak+1 ) and Ex0 ,m denotes expectation with respect to the stochastic process conditioned on X(0) = x0 and n(0) = m. Given the solution pn (x, t|x0 , m, 0) to the CK equation (2.3), we have ∞ ak+1 m τ(2.8) (x ) = dx dt pn (x, t|x0 , m, 0). 0 k Setting qm (x0 , t) =

n=0,1

n=0,1

ak

0

pn (x, t|x0 , m, 0), the backward CK equation takes the form

∂qm ∂ 2 qm (x0 , t) = Dm + A (2.9) mn qn (x0 , t). ∂t ∂x02 n=0,1

The associated boundary conditions are qm (0, t) = 0,

∂qm (L, t) = 0. ∂x0

It follows that τkm evolves according to ∂ 2 τkm (x0 ) Dm + A (2.10) mn τn (x0 ) = −I(ak ,ak+1 ) (x), 2 ∂x0 n=0,1

supplemented by the boundary conditions τkm (0) = 0,

τkm (L) = 0.

4



2.2. Probabilistic formulation

One could determine the mean residence times τkm by explicitly solving the piecewise differ ential equations (2.10). However, this becomes considerably more involved in the gated case, see section 3. Therefore, we will consider an alternative, probabilistic formulation of the above process, which will allow us to apply methods developed in previous work to the analysis of gated residence times [7]. In addition to simplifying the analysis, our approach has a number of other advantages. First, it provides insights into the nature of sample paths that contribute to the residence times. Second, the method can be extended to Brownian particles moving in a √ potential V, for which equation (2.2) becomes dX(t) = V(X)dt + 2Dn dW(t). Although the resulting Chapman–Kolmogorov equation cannot be solved exactly, except for special choices of V, qualitative aspects of the dynamics can be obtained using the probabilistic approach, see for example [4, 5]. For ease of notation we drop the subscript on the initial position x0. Before proceeding, it is useful to recall a few basic definitions from probability theory. Conditional expectations and the tower property. Consider a sample space Ω with σ-algebra

F and probability measure P. In the case of two random variables on the probability space (Ω, F, P), we define the conditional expectation of Y given X by E(Y|X) = yρ(y|X)dy,

where ρ(y|X) is the conditional probability density with respect to X. This definition can be generalized to conditional expectation with respect to a σ-algebra (instead of with respect to a random variable), see [13, 15]. The conditional expectation satisfies E(E(Y|X)) ≡ yρ(y|x)ρ(x)dydx = yρ2 (y, x)dydx = E(Y), where ρ2 is a joint probability density. Using a similar argument, one can also derive the tower property E(E(Y|X1 , X2 )|X1 ) = E(Y|X1 ). Stopping times and the strong Markov property. Let X = {X(t), t ∈ R+ } be a continuous

stochastic process defined on (Ω, F, P). The σ-algebra generated by the stochastic process X up to time t then corresponds to sets of sample paths, realizations or trajectories {X(s), 0 s t}. A stopping time T is a time that depends on the path {X(t), t ∈ R+ }, and is thus a random variable. A defining feature of a stopping time is that one knows at time t whether or not T t , that is, knowledge of the sample path {X(s), s t} is sufficient to determine whether or not T t . It immediately follows that the first passage time (2.5) is a stopping time. Given any stopping time T with respect to X, if the stochastic process Y(t) = X(t + T ) − X(T ) is independent of {X(s), s < T } then X is said to satisfy the strong Markov property. We will make repeated use of the strong Markov property and conditional expectations in the following. Define the first time the particle reaches position y ∈ [a0 , am ] when the jump process is in state n ∈ {0, 1}, sny := inf t > 0 : {X(t) = y} ∩ {n(t) = n} . (2.11) 5



For any stopping time S, we denote the σ-algebra generated by the process {(X(t), n(t))}St=0 until time S by F(S). If x ∈ [a0 , ak ], then by the tower property of conditional expectation and the strong Markov property, we have that τkn (x) = Ex,n [Tk 1s0a

k

1s0a

k